图的不变链
Invariant chains of graphs
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中文总结 AI 辅助
该研究系统探讨图的Inc-不变链的渐近性质，证明独立数、匹配数等组合不变量呈现稳定、线性或拟多项式等刚性渐近行为，揭示受递增对称性支配的图族的强渐近正则性。
中文摘要 AI 辅助
我们启动对图的Inc-不变链的系统研究，这类图是等变诺特性理论中边理想的Inc-不变链的组合对应物。此类链由顶点集不断增长的图构成，其边集与正整数上严格递增映射幺半群的作用相容。我们证明，若干相关组合不变量呈现出刚性渐近行为：独立数最终稳定，独立复形的f-向量和h-向量的每个固定项最终呈线性；团复形的f-向量每个固定项最终呈多项式，h-向量项最终拟多项式；团数和色数最终拟线性，且二者之差最终至多为1；匹配数最终达到最大值⌊n/2⌋；此外，可容许路径和极小路径的最终长度分别至多为3和5，且其最大长度稳定。这些结果揭示了受递增对称性支配的图族中存在强渐近正则性。
英文摘要
We initiate a systematic study of Inc-invariant chains of graphs, the combinatorial counterparts of Inc-invariant chains of edge ideals arising in the theory of equivariant Noetherianity. Such a chain consists of graphs on growing vertex sets whose edge sets are compatible with the action of the monoid of strictly increasing maps on the positive integers. We show that several associated combinatorial invariants exhibit rigid asymptotic behavior. The independence number eventually stabilizes, and every fixed entry of the $f$-vector and the $h$-vector of the independence complex is eventually linear. For clique complexes, every fixed entry of the $f$-vector is eventually polynomial, whereas the entries of the $h$-vector are eventually quasi-polynomial. Moreover, the clique and chromatic numbers are eventually quasi-linear, and their difference is eventually at most one. We also prove that the matching number eventually attains the maximal value $\lfloor n/2\rfloor$. Finally, admissible and minimal paths eventually have lengths at most $3$ and $5$, respectively, and their maximal lengths stabilize. These results reveal strong asymptotic regularity in graph families governed by increasing symmetry.
